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Unformatted text preview: eigenvectors and
, be the real and imaginary
parts of the complex conjugate eigenvectors. The transformation matrix is nonsingular and where
The solution of the initial value problem will involve the matrix exponential .
In this way we compute the matrix exponential of any matrix that is diagonalizable. 2.6 Multiple Eigenvalues
The commutator of
Fact. If and and is . If the commutator is zero then , then and .
.□ Proof. Generalized Eigenspaces
. Recall that eigenvalue
. This can be rewritten as and eigenvector . Suppose has algebraic multiplicity 1. Then the associated eigenspace is satisfy 4
Chapter 2 part B . A space is invariant under the action of
invariant under by the fact above. Suppose
is an eigenvalue of
as if implies . For example, with algebraic multiplicity is . Define the generalized .
The symbol refers to generalized eigenspace but coincides with eigenspace if A nonzero solution to is a generalized eigenvector of . Lemma 2.5 (Invariance). Each of the generalized eigenspaces of...
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- Fall '14